{
  "abstract": "Let $M$ be a rank-one symmetric space of compact or noncompact type and real dimension $m=kn$, where $k\\in\\{1,2,4,8\\}$ is the real dimension of the underlying division algebra. Meng and Wang proved sharp isoperimetric inequalities for reciprocal sums of the first $p$ or $\\ell$ nonzero Neumann eigenvalues, with indices strictly below $m$ in the nonconstant-curvature cases. We keep the whole $m$-dimensional transplanted first eigenspace of an equimeasurable geodesic ball and treat its mass and stiffness forms as a coupled generalized eigenvalue problem. For every bounded connected smooth domain in a noncompact rank-one symmetricspace we prove the full sharp estimate \\[ \\sum_{j=1}^{m}\\frac{1}{μ_j(Ω)} \\ge \\frac{m}{μ_1(B)}, \\] where $B$ is a geodesic ball with $|B|=|Ω|$; equality characterizes geodesic balls. The Hopf-direction contribution is encoded by a positive, unital, trace-preserving projection on symmetric matrices, and the finite-dimensional closure is supplied by a positive-projection Jensen inequality. In compact type, under the localization hypothesis of Meng--Wang, the same full estimate is obtained whenever $\\cot^2 R\\ge k-1$, where $R$ is the radius of the comparison ball, as well as in the isotropic endpoint cases and for Hopf-balanced angular second moment. Consequently the complete $m$-term inequality holds for all complex and quaternionic projective spaces under the original localization assumption, and for the Cayley plane when $R\\le \\arctan(1/\\sqrt7)$. In the remaining admissible Cayley interval we exhibit a finite-dimensional obstruction showing that the affine tangent-matrix bounds alone cannot yield the full trace estimate. This obstruction concerns the proof mechanism and is not a counterexample to the unresolved spectral inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19837",
  "author_total": 1,
  "authors": [
    {
      "name": "Yifeng Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwawtt72qy221.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19837",
  "fulltext_url": null,
  "id": "fwawtt72qy221",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19837",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221977685",
  "pdf_url": "https://arxiv.org/pdf/2610.19837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwawtt72qy221.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces",
  "title_text": "Coupled reciprocal-sum inequalities for Neumann eigenvalues in rank-one symmetric spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/fwawtt72qy221",
  "venue": "arXiv",
  "versions": []
}